Brownian displacement

Treating Brownian motion as a three-dimensional random walk , the mean Brownian displacement x of a particle from its original... 

Perrin (1908) studied the Brownian displacement (and sedimentation equilibrium under gravity see page 35) for fractionated mastic and gamboge suspensions of known particle size, and calculated values for Avogadro s constant varying between 5.5 x 1023 mol1 and 8 x 1023 mol-1. Subsequent experiments of this nature have... 

Table 2.1 Diffusion coefficients and Brownian displacements calculated for uncharged spheres in water at 20°C...

Table 7.1. Root mean square Brownian displacement and distance fallen in one second by unit density particles...

Flow and diffusion in the mobile phase. As noted in Chapter 4, the microscopic flow process in a packed bed is very tortuous, each streampath suffering frequent changes in direction and velocity. A solute molecule carried in this flow will trace out a very uneven path. The randomness of the molecular path will be amplified by Brownian displacements of the molecule from one streampath to another (Figure 11.1). 

The length scale associated with diffusive transport can be characterized in terms of the root-mean-square (rms) Brownian displacement, which is given by (Hinds 1999) ... 

Net Displacement (over a Single Second) Due to Brownian Motion of Spheres of Standard Density Particle Diameter (pm) RMS Brownian Displacement (pm/s)... 

Table 11. Root-mean-square Brownian displacement in Is compared with distance fallen in air in 1 s of unit density particles of different diameters.11 Reproduced with permission from Valberg PA (1985). Determination of retained lung dose from toxicology of inhaled materials. In Handbook of Experimental Pharmacology Vol. 75 (HP Witschi and JD Brain, eds), pp. 57-91. Berlin Springer-Verlag. 1985 Springer-Verlag...

Aspect Diameter (pm) Brownian displacement in 1 s (pm) Distance fallen in 1 s (pm)... 

Brownian motion theory was verified by many scientists (T. Svedberg, A. Westgren, J.Perrin, L.de Broglie and others), who both observed individual particles and followed the diffusion in disperse systems [5]. The influence of various factors, such as the temperature, dispersion medium viscosity, and particle size on the value of the Brownian displacement, was evaluated. It was shown that the Einstein-Smoluchowski theory describes the experimental data adequately and with high precision. 

In quiescent, dilute suspensions, the light fluctuations result essentially from the Brownian displacement of the single particles. Thus, the decay rate F can be traced back to the particles translational diffusion coefficient D, ... 

Particle diameter (urn) Slip correction factor Settling velocity (cm/s) Dilfusion coefficient (cm /s) Mobility [cm/(s dyn)] rms Brownian displacement (cm) ... 

Sehy, J.V., Ackerman, J.J.H., and Neil, J.J., Apparent diffusion of water, ions, and small molecules in the xenopus oocyte is consistent with brownian displacement, Magn. Reson. Med., 48, 42, 2002. 

As expected, the larger the diffusion coefficient, the lower the drag force. Of course, Einstein s diffusion law can be combined with Stokes equation for/ and the resulting equation is called Stokes-Einstein law (Problem 8.1). Together with the equation for the Brownian displacement, it was used by Perrin for early, rather accurate calculations of the Avogadro number. 

Then show that the Brownian displacement of a particle from its original position is given as a function of its radius by the equation ... 

The dynamic friction term gives the correlation between the Brownian displacement in each time interval and the driven motions of the same particle at later times. As was first shown by Mazo(33), these correlations are nonzero and serve to retard diffusion. Correlations between Brownian and driven motions arise because each particle in solution is surrounded by its radial distribution function. The particle s Brownian displacements momentarily carry it off-center relative to the spherically symmetric distribution of neighboring particles. Until the radial distribution relaxes to the new location of the particle, which does not occur instantaneously, the neighboring particles tend to drive the particle of interest back to its initial location. The contributions of dynamic friction to D (34), Ds(35), the drag coefficient fo for motion at constant velocity(36), and r (37) have been obtained. 

What physical forces affect colloid dynamics Three forces acting on neutral colloids are readily identified, namely random thermal forces, hydrodynamic interactions, and direct interactions. The random thermal forces are created by fluctuations in the surrounding medium they cause polymers and colloids to perform Brownian motion. As shown by fluctuation-dissipation theorems, the random forces on different colloid particles are not independent they have cross-correlations. The cross-correlations are described by the hydrodynamic interaction tensors, which determine how the Brownian displacements of nearby colloidal particles are correlated. The hydrodynamic drag experienced by a moving particle, as modified by hydrodynamic interactions with other nearby particles, is also described by a hydrodynamic interaction tensor. 

Hydrodynamic interactions on a long length scale can be measured with two-point rheology, in which fluorescent or other beads are mixed with a polymer solution, and videomicroscopy is used to measure the Brownian displacements AR, of pairs of beads. The cross-correlations AR, ARy) determine the cross-diffusion tensors as a function of the separation between beads. For beads a fraction of a micron in size in polymer solutions and interbead distances out to 100 xm, measurements of Crocker, et al.(25), Gardel, et a/.(26), and Chen, et al.(21) agree the cross-diffusion tensor falls off with distance as /R, and has at least approximately the magnitude expected for the Oseen interaction in these viscous polymer... 

For a particle with velocity u in jc-direction, the travel distance is Ajc = uAt during the pulse interval of At. The relative error due to the Brownian motion during this time interval can be expressed as the ratio of the rms of Brownian displacement to the average motion, that is. 

Membrane subunits (Fig. 16b) are embedded in a medium of viscosity much greater than that of water. An exact relationship between time t (second), particle radius r (cm), viscosity 17 (poise), and Brownian displacement A (cm) is given by... 

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